quantum phase transitions and disorder: rare regions, gri–ths … · 2005. 1. 19. · quantum...
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Quantum phase transitions and disorder:Rare regions, Griffiths effects, and smearing
Thomas VojtaDepartment of Physics, University of Missouri-Rolla
• Phase transitions and quantum phase transitions• Quenched disorder and critical behavior: the common lore
• Rare regions and Griffiths effects• Smeared phase transitions
• An attempt of a classification
Santa Barbara, Jan 18, 2005
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Acknowledgements
at UMR
Rastko SknepnekMark DickisonBernard Fendler
collaboration
Jorg SchmalianMatthias Vojta
Funding:
University of Missouri Research BoardNational Science Foundation CAREER Award
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Classical and quantum phase transitions
classical phase transitions: at non-zero temperature, asymptotic criticalbehavior dominated by classical physics (thermal fluctuations)
quantum phase transitions: at zero temperature as function of pressure,magnetic field, chemical composition, ..., driven by quantum fluctuations
paramagnet
ferromagnet
0 20 40 60 80transversal magnetic field (kOe)
0.0
0.5
1.0
1.5
2.0
Bc
LiHoF4
phase diagrams of LiHoF4 (Bitko et al. 96) and MnSi (Pfleiderer et al. 97)
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Imaginary time and quantum to classical mapping
Classical partition function: statics and dynamics decouple
Z =∫
dpdq e−βH(p,q) =∫
dp e−βT (p)∫
dq e−βU(q) ∼ ∫dq e−βU(q)
Quantum partition function:
Z = Tre−βH = limN→∞(e−βT/Ne−βU/N)N =∫
D[q(τ)] eS[q(τ)]
imaginary time τ acts as additional dimensionat T = 0, the extension in this direction becomes infinite
Caveats:• mapping holds for thermodynamics only• resulting classical system can be unusual and anisotropic (z 6= 1)• extra complications with no classical counterpart may arise, e.g., Berry phases
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• Phase transitions and quantum phase transitions
• Quenched disorder and critical behavior: the common lore
• Rare regions and quantum Griffiths effects
• Smeared phase transitions
• An attempt of a classification
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Weak disorder and Harris criterion
weak disorder: impurities lead to spatial variation of coupling strength g(x)Theory: random mass disorder Experiment: “good disorder” ???
Harris criterion: variation of average local gc(x) in correlation volume must besmaller than distance from global gc
variation of average gc in volume ξd
∆〈gc(x)〉 ∼ ξ−d/2
distance from global critical point t ∼ ξ−1/ν
∆〈gc(x)〉 < t ⇒ dν > 2
ξ
+gC(1),
+gC(4),
+gC(2),
+gC(3),
• if clean critical point fulfills Harris criterion ⇒ stable against disorder• inhomogeneities vanish at large length scales• macroscopic observables are self-averaging• example: 3D classical Heisenberg magnet: ν = 0.698
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Finite-disorder critical points
if critical point violates Harris criterion ⇒ unstable against disorder
Common lore:
• system goes to new different critical point which fulfills dν > 2• inhomogeneities remain finite at all length scales (”finite disorder”)• macroscopic observables are not self-averaging• example: 3D classical Ising magnet: clean ν = 0.627 ⇒ dirty ν = 0.684
Distribution of criticalsusceptibilities of 3D dilute Isingmodel(Wiseman + Domany 98)
0.0 0.5 1.0 1.5 2.0 2.5χ(Tc,L)/[χ(Tc,L)]
0.00
0.02
0.04
0.06
0.08
0.10
freq
uenc
y
10.0 100.0L
0.10
0.12
0.14
0.16
0.18
0.20
Rχ
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Disorder and quantum phase transitions
Disorder is quenched:
• impurities are time-independent
• disorder is perfectly correlated in imaginarytime direction
⇒ correlations increase the effects of disorder(”it is harder to average out fluctuations”)
x
τ
Disorder generically has stronger effects on quantum phase transitionsthan on classical transitions
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Random quantum Ising model
H = −∑
〈i,j〉Jijσ
zi σ
zj −
∑
i
hiσxi
nearest neighbor interactions Jij and transverse fields hi both random
Exact solution in 1+1 dimensions:Ma-Dasgupta-Hu-Fisher real space renormalization group
• in each step, integrate out largest energy among all Jij and hi
• cluster aggregation/annihilation procedure• becomes exact in the limit of large disorder
Infinite-disorder critical point:
• under renormalization the disorder increases without limit• relative width of the distributions of Jij, hi diverges
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Infinite-disorder critical point
• extremely slow dynamics log ξτ ∼ ξµ (activated scaling)• distributions of macroscopic observables become infinitely broad• average and typical values can be drastically different
correlations: − log Gtyp ∼ rψ Gav ∼ r−η
• averages are dominated by rare events
Probability distribution
of the end-to-end
correlations in a random
quantum Ising chain
(Fisher + Young 98)
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• Phase transitions and quantum phase transitions
• Quenched disorder and critical behavior: the common lore
• Rare regions and Griffiths effects
• Smeared phase transitions
• An attempt of a classification
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Griffiths effects in a classical dilute ferromagnet
• critical temperature Tc is reducedcompared to clean value Tc0
• for Tc < T < Tc0:no global order but local order on rareregions devoid of impurities
• probability: w(L) ∼ e−cLd:
rare regions have slow dynamics⇒ singular free energy everywhere in the Griffiths region (Tc < T < Tc0)
Classical Griffiths effects are generically weak and essentially unobservable
contribution to susceptibility: χRR ∼∫
dL e−cLdLγ/ν = finite
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Quantum Griffiths effects
rare regions at a QPT are finite inspace but infinite in imaginary time
fluctuations of the rare regions areeven slower than in classical case ⇒Griffiths singularities are enhanced
t
rare region at a quantum phase
transition
Random quantum Ising systems
local susceptibility (inverse energy gap) of rare region: χloc ∼ ∆−1 ∼ eaLd
χRR ∼∫
dL e−cLdeaLd
can diverge inside Griffiths region
finite temperatures:χRR ∼ T d/z′−1 (z′ is continuously varying exponent)
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• Phase transitions and quantum phase transitions
• Quenched disorder and critical behavior: the common lore
• Rare regions and Griffiths effects
• Smeared phase transitions
• An attempt of a classification
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Rare regions at quantum phase transitions with overdampeddynamics
itinerant Ising quantum antiferromagnet
magnetic fluctuations are damped due to coupling to electrons
Γ(q, ωn) = t + q2 + |ωn|
in imaginary time: long-range power-law interaction ∼ 1/(τ − τ ′)2
1D Ising model with 1/r2 interaction is known to have an ordered phase
⇒ isolated rare region can develop a static magnetization, i.e.,large islands do not tunnel (c.f. Millis, Morr, Schmalian and Castro-Neto)
⇒ conventional quantum Griffiths behavior does not existmagnetization develops gradually on independent rare regions
quantum phase transition is smeared by disorder
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Phase diagram at a smeared transition
T
x
Magneticphase
exponentialtail
possible realization: ferromagnetic quantum phase transition in NixPd1−x
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Universality of the smearing scenario
Condition for disorder-induced smearing:
isolated rare region can develop a static order parameter⇒ rare region has to be above lower critical dimension
Examples:
• quantum phase transitions of itinerant electrons(disorder correlations in imaginary time + long-range interaction 1/τ2)
• classical Ising magnets with planar defects(disorder correlations in 2 dimensions)
• classical non-equilibrium phase transitions in the directed percolationuniversality class with extended defects(disorder correlations in at least one dimension)
Disorder-induced smearing of a phase transition is aubiquitous phenomenon
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Isolated islands – Lifshitz tail arguments
probability to find rare region of size L devoid of defects: w ∼ e−cLd
region has transition at distance tc(L) < 0 from the clean critical pointfinite size scaling: |tc(L)| ∼ L−φ (φ = clean shift exponent)
Consequently:probability to find a region which becomes critical at tc:
w(tc) ∼ exp(−B |tc|−d/φ)
total magnetization at coupling t is given by the sum over all rare regions havingtc > t:
m(t) ∼ exp(−B |t|−d/φ) (t → 0−)
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Computer simulation of a model system
Classical Ising model in 2+1 dimensions
H = −1Lτ
∑
〈x,y〉,τ,τ ′Sx,τSy,τ ′ −
1Lτ
∑
x,τ,τ ′JxSx,τSx,τ ′
Jx: binary random variable, P (J) = (1− c) δ(J − 1) + c δ(J)totally correlated in the time-like direction
• short-range interactions in the two space-like directions• infinite-range interaction in the time-like direction
(static magnetization on the rare regions is retained, but time directioncan be treated exactly, permitting large sizes)
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Smeared transition in the infinite-range model
4.75 4.8 4.85 4.9 4.95 5 5.05T
0
50
100
χ
0
0.05
0.1
0.15
0.2
0.25
m
susceptibilitymagnetization
4.87 4.88 4.89 4.9T
0
50
100
χ
4.87 4.88 4.89 4.9T
0.00
0.01
0.02
0.03
m
L=100L=200L=400L=1000
Magnetization + susceptibilityof the infinite-range model
seeming transition close toT = 4.88
phase transition is smeared
(m and χ are independent of L)
Lifshitz magnetization tailtowards disordered phaselog(m) ∼ − 1/(Tc0 − T )
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Local magnetization distribution
0 50 100 150 200 250 300 350 400x 0
50100
150200
250300
350400
y
00.020.040.060.080.1
0.120.140.160.18
m
-25 -20 -15 -10 -5 0log(m)
10-4
10-3
10-2
10-1
100
P[lo
g(m
)]
4.914.904.894.88
4.874.864.854.84
Local magnetization in thetail region (T = 4.8875)
global magnetization startsto form on isolated islands
very inhomogeneous system
Distribution of the localmagnetization values
very broad, even onlogarithmic scale
ln(mtyp) ∼ 〈m〉−1/2
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• Phase transitions and quantum phase transitions
• Quenched disorder and critical behavior: the common lore
• Rare regions and Griffiths effects
• Smeared phase transitions
• An attempt of a classification
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Classification of dirty phase transitions according toimportance of rare regions
Dimensionality Griffiths effects Dirty critical point Examplesof rare regions (classical PT, QPT)
dRR < d−c weak exponential conv. finite disorder class. magnet with point defects
dilute bilayer Heisenberg model
dRR = d−c strong power-law infinite randomness Ising model with linear defects
random quantum Ising model
itin. quantum Heisenberg magnet?
dRR > d−c RR become static smeared transition Ising model with planar defects
itinerant quantum Ising magnet
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Dimer-diluted 2d Heisenberg quantum antiferromagnet
H = J‖∑〈i,j〉
a=1,2
εiεjSi,a · Sj,a + J⊥∑
i
εiSi,1 · Si,2,
Large scale Monte-Carlo simulations:
conventional finite-disorder critical point with power-law scalingcritical exponents are universal, dynamical exponent z = 1.31(after accounting for corrections to scaling)
0 0.1 0.2 0.3 0.4p
0
0.5
1
1.5
2
2.5
J ⊥ /
J ||
AF ordered
disordered
J
J2
z
5 10 20 50 100L
1
1.2
1.5
2
2.5
3
4
Ltm
ax/L
p=1/3p=2/7p=1/5p=1/8
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Conclusions
• even weak disorder can have surprisingly strong effects on a quantum phasetransition
• rare regions play a much bigger role quantum phase transitions than aclassical transitions
• effective dimensionality of rare regions determines overall phenomenologyof phase transitions in disordered systems
• Ising systems with overdamped dynamics: sharp phase transition is destroyedby smearing because static order forms on rare spatial regions
• at a smeared transition, system is extremely inhomogeneous, even on alogarithmic scale
Griffiths effects at quantum phase transitions leads to a rich variety ofnew and exotic phenomena